Graph f(x)=x^2-8x+15

Solve Math

Algebra

$f (x) = x^{2} - 8 x + 15$

Find the properties of the given parabola.

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Rewrite the equation in vertex form.

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Complete the square for $x^{2} - 8 x + 15$ .

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Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1, b = - 8, c = 15$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = - \frac{8}{2 (1)}$

Simplify the right side.

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Cancel the common factor of $8$ and $2$ .

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Factor $2$ out of $8$ .

$d = - \frac{2 \cdot 4}{2 \cdot 1}$

Cancel the common factors.

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Factor $2$ out of $2 \cdot 1$ .

$d = - \frac{2 \cdot 4}{2 (1)}$

Cancel the common factor.

$d = - \frac{2 \cdot 4}{2 \cdot 1}$

Rewrite the expression.

$d = - \frac{4}{1}$

Divide $4$ by $1$ .

$d = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$d = - 4$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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Raise $- 8$ to the power of $2$ .

$e = 15 - \frac{64}{4 (1)}$

Multiply $4$ by $1$ .

$e = 15 - \frac{64}{4}$

Divide $64$ by $4$ .

$e = 15 - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$e = 15 - 16$

Subtract $16$ from $15$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $a {(x + d)}^{2} + e$ .

${(x - 4)}^{2} - 1$

Set $y$ equal to the new right side.

$y = {(x - 4)}^{2} - 1$

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 1$

$h = 4$

$k = - 1$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(4, - 1)$

Find $p$ , the distance from the vertex to the focus.

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Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 1}$

Cancel the common factor of $1$ .

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Cancel the common factor.

$\frac{1}{4 \cdot 1}$

Rewrite the expression.

$\frac{1}{4}$

Find the focus.

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The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(4, - \frac{3}{4})$

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 4$

Find the directrix.

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The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{5}{4}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: $(4, - 1)$

Focus: $(4, - \frac{3}{4})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{5}{4}$

Direction: Opens Up

Vertex: $(4, - 1)$

Focus: $(4, - \frac{3}{4})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{5}{4}$

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Replace the variable $x$ with $3$ in the expression.

$f (3) = {(3)}^{2} - 8 \cdot 3 + 15$

Simplify the result.

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Simplify each term.

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Raise $3$ to the power of $2$ .

$f (3) = 9 - 8 \cdot 3 + 15$

Multiply $- 8$ by $3$ .

$f (3) = 9 - 24 + 15$

Simplify by adding and subtracting.

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Subtract $24$ from $9$ .

$f (3) = - 15 + 15$

Add $- 15$ and $15$ .

$f (3) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 3$ is $0$ .

$y = 0$

Replace the variable $x$ with $2$ in the expression.

$f (2) = {(2)}^{2} - 8 \cdot 2 + 15$

Simplify the result.

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Simplify each term.

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Raise $2$ to the power of $2$ .

$f (2) = 4 - 8 \cdot 2 + 15$

Multiply $- 8$ by $2$ .

$f (2) = 4 - 16 + 15$

Simplify by adding and subtracting.

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Subtract $16$ from $4$ .

$f (2) = - 12 + 15$

Add $- 12$ and $15$ .

$f (2) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = 2$ is $3$ .

$y = 3$

Replace the variable $x$ with $5$ in the expression.

$f (5) = {(5)}^{2} - 8 \cdot 5 + 15$

Simplify the result.

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Simplify each term.

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Raise $5$ to the power of $2$ .

$f (5) = 25 - 8 \cdot 5 + 15$

Multiply $- 8$ by $5$ .

$f (5) = 25 - 40 + 15$

Simplify by adding and subtracting.

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Subtract $40$ from $25$ .

$f (5) = - 15 + 15$

Add $- 15$ and $15$ .

$f (5) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 5$ is $0$ .

$y = 0$

Replace the variable $x$ with $6$ in the expression.

$f (6) = {(6)}^{2} - 8 \cdot 6 + 15$

Simplify the result.

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Simplify each term.

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Raise $6$ to the power of $2$ .

$f (6) = 36 - 8 \cdot 6 + 15$

Multiply $- 8$ by $6$ .

$f (6) = 36 - 48 + 15$

Simplify by adding and subtracting.

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Subtract $48$ from $36$ .

$f (6) = - 12 + 15$

Add $- 12$ and $15$ .

$f (6) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = 6$ is $3$ .

$y = 3$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 3 \\ 3 & 0 \\ 4 & - 1 \\ 5 & 0 \\ 6 & 3 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, - 1)$

Focus: $(4, - \frac{3}{4})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{5}{4}$

$\begin{array}{c} x & y \\ 2 & 3 \\ 3 & 0 \\ 4 & - 1 \\ 5 & 0 \\ 6 & 3 \end{array}$

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Name

Name	four hundred seventy-five million two hundred forty-five thousand forty-four

Interesting facts

475245044 has 16 divisors, whose sum is 1115792928
The reverse of 475245044 is 440542574
Previous prime number is 23

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 35
Digital Root 8

Name

Name	four hundred five million five hundred sixteen thousand seven hundred eighty

Interesting facts

405516780 has 64 divisors, whose sum is 1056097440
The reverse of 405516780 is 087615504
Previous prime number is 53

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 36
Digital Root 9

Name

Name	one billion six hundred eight million fifty-seven thousand seven hundred sixty-six

Interesting facts

1608057766 has 16 divisors, whose sum is 2775175200
The reverse of 1608057766 is 6677508061
Previous prime number is 149

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 46
Digital Root 1

Factor x^2-4

Evaluate square root of -17

Solve Using the Square Root Property (x-3)^2=5

Evaluate 2 square root of 4

Add (2-8i)+(5-i)